where the superscript \(\gamma \)  labels the orientation of the bond, \(R_{\alpha \beta }^\gamma = -\frac{i}{\sqrt{2}}(\sigma ^\alpha + \sigma ^\beta )\)  is a bond-dependent rotation matrix[1]}, \(\bar{f_i}=(f_{i\downarrow }^\dagger , -f_{i\uparrow }^\dagger )^T.\)  \(\Delta _{0,1,2,3}^\gamma \)  and \(t_{0,1,2,3}^\gamma \)  are the mean field parameters (complex numbers) representing the fermion pairing and hopping terms, respectively. Owing to the \(C_3\)  rotation symmetry the values of \(t^\gamma \)  and \(\Delta ^\gamma \)  are independent on the bond orientation, so we will only consider the \(z\) -bonds and omit the bond index in later discussion. The self consistent mean-field equations for these parameters read
\( t_1&=&g_1 {2\over N}\sum _{\langle ij\rangle \in z} \langle f_i^\dagger R_{xy} f_j \rangle ^*, \\t_{0,2} &=& g_{0,2}{2\over N}\sum _{\langle ij\rangle \in z} \left[\langle f_i^\dagger \sigma ^{x} R_{xy} f_j \rangle ^* \pm \langle f_i^\dagger \sigma ^{y} R_{xy} f_j \rangle ^*\right], \nonumber \\t_3 &=& g_3{2\over N}\sum _{\langle ij\rangle \in z} \langle f_i^\dagger \sigma ^z R_{xy} f_j\rangle ^*, \\\Delta _1 &=& g_1^{^{\prime }}{2\over N}\sum _{\langle ij\rangle \in z} \langle f_i^\dagger R_{xy} \bar{f}_j \rangle ^*, \nonumber \\\Delta _{0,2} &=& g_{0,2}^{^{\prime }}{2\over N}\sum _{\langle ij\rangle \in z} \left[\langle f_i^\dagger \sigma ^{x} R_{xy} \bar{f}_j \rangle ^* \pm \langle f_i^\dagger \sigma ^{y} R_{xy} f_j \rangle ^*\right], \nonumber \\ \Delta _3 &=& g_3^{^{\prime }} {2\over N}\sum _{\langle ij\rangle \in z} \langle f_i^\dagger \sigma ^z R_{xy} \bar{f}_j\rangle ^*,\) 
